Solve the Inequality for x -x^2-2>=8x+5

$- x^{2} - 2 \geq 8 x + 5$

Subtract $8 x$ from both sides of the inequality.

$- x^{2} - 2 - 8 x \geq 5$

Convert the inequality to an equation.

$- x^{2} - 2 - 8 x = 5$

Subtract $5$ from both sides of the equation.

$- x^{2} - 2 - 8 x - 5 = 0$

Subtract $5$ from $- 2$ .

$- x^{2} - 8 x - 7 = 0$

Factor the left side of the equation.

Factor $- 1$ out of $- x^{2} - 8 x - 7$ .

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) - 8 x - 7 = 0$

Factor $- 1$ out of $- 8 x$ .

$- (x^{2}) - (8 x) - 7 = 0$

Rewrite $- 7$ as $- 1 (7)$ .

$- (x^{2}) - (8 x) - 1 \cdot 7 = 0$

Factor $- 1$ out of $- (x^{2}) - (8 x)$ .

$- (x^{2} + 8 x) - 1 \cdot 7 = 0$

Factor $- 1$ out of $- (x^{2} + 8 x) - 1 (7)$ .

$- (x^{2} + 8 x + 7) = 0$

Factor.

Factor $x^{2} + 8 x + 7$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $7$ and whose sum is $8$ .

$1, 7$

Write the factored form using these integers.

$- ((x + 1) (x + 7)) = 0$

Remove unnecessary parentheses.

$- (x + 1) (x + 7) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$x + 7 = 0$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $x + 7$ equal to $0$ and solve for $x$ .

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

The final solution is all the values that make $- (x + 1) (x + 7) = 0$ true.

$x = - 1, - 7$

Use each root to create test intervals.

$x < - 7$

$- 7 < x < - 1$

$x > - 1$

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < - 7$ to see if it makes the inequality true.

Choose a value on the interval $x < - 7$ and see if this value makes the original inequality true.

$x = - 10$

Replace $x$ with $- 10$ in the original inequality.

$- {(- 10)}^{2} - 2 \geq 8 (- 10) + 5$

The left side $- 102$ is less than the right side $- 75$ , which means that the given statement is false.

False

Test a value on the interval $- 7 < x < - 1$ to see if it makes the inequality true.

Choose a value on the interval $- 7 < x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Replace $x$ with $- 4$ in the original inequality.

$- {(- 4)}^{2} - 2 \geq 8 (- 4) + 5$

The left side $- 18$ is greater than the right side $- 27$ , which means that the given statement is always true.

True

Test a value on the interval $x > - 1$ to see if it makes the inequality true.

Choose a value on the interval $x > - 1$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

$- {(0)}^{2} - 2 \geq 8 (0) + 5$

The left side $- 2$ is less than the right side $5$ , which means that the given statement is false.

False

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 7$ False

$- 7 < x < - 1$ True

$x > - 1$ False

$x < - 7$ False

$- 7 < x < - 1$ True

$x > - 1$ False

The solution consists of all of the true intervals.

$- 7 \leq x \leq - 1$

The result can be shown in multiple forms.

Inequality Form:

$- 7 \leq x \leq - 1$

Interval Notation:

$[- 7, - 1]$

Solve the Inequality for x -x^2-2>=8x+5

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